Constructing extensions of ultraweakly continuous linear functionals

Let R be a linear subset of the space B(H) of bounded operators on a Hilber t space H with an orthonormal basis. It is shown constructively that if the unit ball of R is weak-operator totally bounded, then an ultraweakly conti nuous linear functional on A extends to one on B(H), and the extended funct ional has the form T bar right arrow Sigma (infinity)(n=1) [Tx(n), y(n)], w here Sigma (infinity)(n-1) parallel to xn parallel to (2) and Sigma (infini ty)(n=1) parallel toy(n)parallel to (2) are convergent series in R. (C) 200 0 Academic Press.